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Define \(f : \mathbb{R} \to \mathbb{R}\) by \(f(x) = x^2 \sin(1/x)\) for \(x \ne 0\) (admit \(\sin\) from Standard functions) and \(f(0) = 0\). Show that \(f\) is differentiable at \(0\) with \(f'(0) = 0\). Is \(f\) of class \(C^1\) on \(\mathbb{R}\)? (Hint: study \(\lim_{x \to 0} f'(x)\).)
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